Gram-like matrix preserving extensions and completions of noncommutative polynomials

TL;DR

This paper introduces a novel approach to extend noncommutative polynomials to sums of Hermitian squares while preserving their Gram matrices, using linear algebra and positive semidefinite completion techniques.

Contribution

It proposes the concept of Gram-like matrices and develops linear algebraic methods for extending noncommutative polynomials without perturbing their Gram matrices.

Findings

Developed criteria using chordal graphs for positive semidefinite completion.

Provided methods to extend noncommutative polynomials to sums of Hermitian squares.

Linked Gram matrix structures to algebraic and graph-theoretic properties.

Abstract

Given a positive noncommutative polynomial $f$, equivalently a sum of Hermitian squares (SOHS), there exists a positive semidefinite Gram matrix that encrypts all the structural essence of $f$. There are no available methods for extending a noncommutative polynomial to a SOHS keeping the Gram matrices unperturbed. As a remedy, we introduce an equally significant notion of Gram-like matrices and provide linear algebraic techniques to get the desired extensions. We further use positive semidefinite completion problem to get SOHS and provide criteria in terms of chordal graphs and 2-regular projective algebraic sets.